\section{Weak Nuclear Statistical Equilibrium and the Production of Neutron-Rich Iron-Group Isotopes}
\label{appendix:wse}

This work is by Tianhong Yu and Bradley S. Meyer [PoS(NIC XII)213].

\subsection{Abstract}

Calcium-48, $^{50}$Ti, and $^{54}$Cr are neutron-rich iron-group
isotopes that show roughly correlated excesses and deficits in certain 
calcium-aluminum-rich inclusions (CAIs) in primitive meteorites.
These isotopes are produced in high-temperature, low-entropy-per-nucleon
environments such that the nuclear population are governed by a 
quasi-statistical equilibrium with too many heavy nuclei compared to nuclear
statistical equilibrium.
Such environments are present in dense
thermonuclear (Type Ia) supernovae.
Production of these isotopes also
requires an electron fraction $Y_{\rm e}$ approximately equal to 0.42, which is
set by electron captures during the explosion. We use NucNet Tools, an
open-source suite of tools for nucleosynthesis, to study 
nucleosynthesis in high-density, low-entropy environments appropriate for 
Type Ia supernovae and follow the neutronization of the matter by weak
interactions. We study how the nuclear populations evolve towards and into 
dynamical weak statistical equilibrium.
%Interestingly, for temperature
%in the range of $\rm{T_9}$ = 4-6, the evolution to weak statistical equilibrium
%is via a quasi-equilibrium, not a nuclear statistical equilibrium, because
%the timescale to change the number of heavy nuclei is comparable to the 
%timescale to change $Y_{\rm e}$.
Realistic expansion timescales for dense Type Ia supernova matter do not
allow the material to reach dynamical weak statistical equilibrium.
Neverthless, such expansions are able
to generate low enough $Y_{\rm e}$ to produce $^{48}$Ca.  The
conditions in these expansions are extreme and thus 
probably rare in Galactic history, but, when they do occur, they 
produce copious amounts of $^{48}$Ca and the other neutron-rich species. For 
this reason, it is likely that the abundance of these isotopes in interstellar
dust is quite heterogeneous. Because the CAIs formed from interstellar
dust precursors, they inherited this heterogeneity.

\subsection{Introduction}

Primitive Solar System hibonites
and FUN (Fractionated and Unknown Nuclear effects) CAIs
(calcium-aluminum-rich inclusions) show correlated excesses and deficits
in neutron-rich iron-group elements (e.g., \cite{2006mess.book...69M}).
These isotopes are robustly produced in low-entropy expansions of
neutron-rich matter \cite{1996ApJ...462..825M}, and such expansions
likely occur in rare, dense, thermonuclear (Type Ia) supernovae
\cite{1997ApJ...476..801W}.

In this brief paper we explore several low-entropy expansions and
follow the neutronization that occurs due to weak interactions on free
nucleons and nuclei.  We compare network calculations to corresponding
calculations of dynamical weak statistical equilibrium to study the
extent to which the network calculations attain this equilibrium.  For
realistic supernova timescales, the expansions do not attain
dynamical weak statistical equilibrium, but they do reach a degree
of neutron richness sufficient to make neutron-rich iron-group isotopes
like $^{48}$Ca, $^{50}$Ti, and $^{54}$Cr.

\subsection{Dynamical Weak Nuclear Statistical Equilibrium}

Given sufficient time, a system undergoing nucleosynthesis can evolve
until the nuclear species are in full equilibrium under assembly from
free nucleons.
This is the condition of nuclear statistical equilibrium (NSE), for which
the chemical potential for a nuclear species with atomic number $Z$ and mass
number $A$ is related to that for free neutrons ($\mu_n$) and free protons
($\mu_p$) by
\begin{equation}
\mu(Z,A) = Z \mu_p + \left( A - Z \right) \mu_n.
\label{eq:nse}
\end{equation}
A related equilibrium is quasi-statistical equilibrium (QSE) in which
nuclei are in equilibrium under exchange of nucleons, but the total number
of heavy nuclei ($Z > 2$) is not that required by NSE because the
three-body reactions assembling heavy nuclei are too slow
\cite{1996ApJ...462..825M}.  In this case, the chemical potential for
a heavy nucleus is given by
\begin{equation}
\mu(Z,A) = \mu_h + Z \mu_p + \left( A - Z \right) \mu_n.
\label{eq:qse}
\end{equation}
where $\mu_h$ is the chemical potential of the heavy nuclei as a whole.

The NSE and QSE described above consider the total neutron-to-proton
ratio to be fixed.  Weak interactions can change this quantity so that
the system can evolve to weak equilibrium.
If neutrinos are trapped in the material,
the system can evolve to weak nuclear
statistical equilibrium (here called WSE) such that there is an equilibrium
under the interchange of neutrons and protons via weak reactions.  In this
case there is the additional condition relating the chemical potentials of
protons, electrons, neutrons, and electron-type neutrinos:
\begin{equation}
\mu_p + \mu_e = \mu_n + \mu_{\nu_e}
\label{eq:wse}
\end{equation}
In WSE, nuclear abundances per nucleon $Y(Z,A)$ evolve until their chemical
potentials satisfy Eq. (\ref{eq:nse}) and the electron-to-nucleon ratio
$Y_{\rm e}$ satisfies Eq. (\ref{eq:wse}).

Densities are typically not sufficient in white dwarf star cores to trap
neutrinos, however.
In such matter, nuclear populations evolve not to WSE but rather
to a dynamical weak nuclear statistical equilibrium (here called dWSE)
\cite{2010A&A...522A..25A}.
In this case,
the nuclear abundances obey Eq. (\ref{eq:nse}), but the constraint on the
electron-to-nucleon ratio $Y_{\rm e}$ is instead given by the condition that
the time rate of change of $Y_{\rm e} = 0$.  Because of the extremely low
abundance of positrons in degenerate material,
positron capture and $\beta^+$ decay are both small in white-dwarf star
matter so that
this dWSE arises when the total electron capture rate in the matter
(which decreases $Y_{\rm e}$) equals the total $\beta^-$ decay rate (which
increases $Y_{\rm e}$).

\subsection{Network Calculation}

We study the nucleosynthesis of matter expanding from high temperature and
density using NucNet Tools, an open-source suite of tools we have written
to study nucleosynthesis \cite{nnt}.  Our network calculations use reaction
rates from the Joint Institute for Nuclear Astrophysics database
at http://www.jinaweb.org.  We supplement these rates with weak-interaction
rates from \\
\cite{2001ADNDT..79....1L}.  For nuclei that do not yet have
microscopic weak rate estimates available, we use the approximate rate
formulation in \cite{2010A&A...522A..25A}.

\begin{figure}[ht!]
\begin{center}
  \includegraphics[width=\figuresize\textwidth]{figures/wse_figures/ye_t9}
  \caption{The evolution of the electron-to-nucleon ratio
$Y_{\rm e}$ as a function of $T_9 = T / 10^9$ K in
expansions of various density e-folding timescale $\tau$.  Also shown as
the red curve is the dynamical weak statistical equilibrium (dWSE) $Y_{\rm e}$ for the
corresponding density and temperature.  The slow expansions come close to
attaining dWSE at high temperature and density but diverge from dWSE at lower
temperature when the weak rates decrease dramatically.  The faster expansions
never attain dWSE.}
\label{fig:ye_t9}
\end{center}
\end{figure}

We ran a set of calculations that began with 50\% by mass of $^{12}$C and
50\% by mass $^{16}$O.  The matter began at a mass density 
$\rho$ of $9 \times 10^9$ g cm$^{-3}$
and a temperature of $T_9 = T/10^9$K $=10$.  We considered the density to
expand exponentially with time $t$ such that $\rho(t) = \rho(0) \exp(-t/\tau)$,
where $\tau$ is the density e-folding timescale.  We considered the entropy
of the matter to be dominated by relativistic particles such that
$\rho \propto T_9^3$.  We also ran some more detailed models that account
for energy generation that we will describe in a future paper.
These models show that $\rho \propto T_9^3$ is a reasonable parameterization
for these studies.

Fig. \ref{fig:ye_t9} shows the time evolution of the electron-to-nucleon
ratio $Y_{\rm e}$ for $\tau = 0.1, 1, 10,$ and $1000$ seconds.  Also shown on this
curve is the instantaneous dWSE value of $Y_{\rm e}$.  This is the
electron-to-nucleon ratio the matter would evolve to if the expansion
were arrested and the matter allowed to evolve at the fixed temperature
and density at that point in the expansion.  This quantity is computed
by finding the value of $Y_{\rm e}$ at that temperature and density such
that the NSE has a rate of change of $Y_{\rm e}$ equal to zero.
The dWSE $Y_{\rm e}$ is low ($Y_{\rm e} \approx
0.38$) early in the calculation because the density and, hence, the
electron chemical potential is high.  This favors electron capture
and drives the dWSE $Y_{\rm e}$ down.  As the matter expands and cools, the
density drops.  This lowers the electron chemical potential, which lowers
the typical electron-capture rate and increases the dWSE $Y_{\rm e}$.

As Fig. \ref{fig:ye_t9} shows, slow expansion (large $\tau$) allows the
system $Y_{\rm e}$ to keep pace with the dWSE $Y_{\rm e}$ better than the faster expansions.
For $\tau = 1000$ s, the matter attains dWSE at $T_9 = 10$ and evolves
in dWSE until the network diverges from dWSE near $T_9 = 5$ and freezes out
at about $T_9 = 4$.  In contrast,
the faster expansions experience a drop from the initial $Y_{\rm e} = 0.5$ early
due to electron capture but freezeout near their final $Y_{\rm e}$ at about $T_9 = 8$.

\begin{figure}[h]
\begin{center}
  \includegraphics[width=\figuresize\textwidth]{figures/wse_figures/yedot_t}
  \caption{The total electron-capture and $\beta^-$ decay rates as a
function of time during the
fixed temperature and density calculations.  The system attains dWSE
when the two rates become equal.  Note that the lower density calculation
takes a longer time to reach dWSE because the overall rates are lower.}
\label{fig:yedot_t}
\end{center}
\end{figure}

To study the evolution of $Y_{\rm e}$ during these calculations,
we made two calculations of matter with initial mass fractions
of 50\% $^{12}$C and 50\% $^{16}$O evolving at fixed temperature and
density drawn from points in the expansions shown in Fig. \ref{fig:ye_t9}.
The first calculation is for $T_9 = 8$ and
\mbox{$\rho = 4.6 \times 10^9$ g cm$^{-3}$}.
The left panel of Fig. \ref{fig:yedot_t} shows the total electron capture
and total $\beta^-$ decay rate
during this calculation.  Initially electron capture dominates beta decay
and causes the $Y_{\rm e}$ to drop.
As the material becomes neutron rich, however, $\beta^-$ decay becomes
important.  After \mbox{$\sim 10$ s}, the total electron capture rate equals
the total $\beta^-$ decay rate and the matter attains dWSE.
This explains why both the $\tau = 10$ s and $\tau = 1000$ s expansions
are near dWSE at $T_9 = 8$ -- the expansion timescale is longer than or
comparable to the timescale to attain dWSE.

The second calculation is for $T_9 = 5.5$ and $\rho = 1.5 \times 10^9$ g cm$^{-3}$.
The right panel of Fig. \ref{fig:yedot_t} shows the total electron
capture and $\beta^-$ decay rates for this calculation.  This figure
is similar to that for the previous calculation.  The difference is that
the overall rates are smaller because of the lower density (and electron
chemical potential) which results in a longer
timescale for evolution to dWSE (larger than
$\sim 100$ s).  The long timescale for
evolution to dWSE means that, even for the \mbox{$\tau = 1000$ s} expansion,
the network has difficulty keeping pace with the changing
dWSE $Y_{\rm e}$ at this temperature and density. 

\begin{figure}
\begin{center}
  \includegraphics[width=\figuresize\textwidth]{figures/wse_figures/yedot_ye}
  \caption{The total electron-capture and $\beta^-$ decay rates as a function
of the material's electron-to-nucleon ratio $Y_{\rm e}$ during the
fixed temperature and density calculations.  Initially the electron captures
dominate $\beta^-$ decay.  As the material becomes more neutron rich,
the $\beta^-$ decay rates increase and the electron-capture rates decrease.
When they become equal, the rate of change of $Y_{\rm e}$ becomes zero and
the material reaches dWSE.} 
\label{fig:yedot_ye}
\end{center}
\end{figure}

Fig. \ref{fig:yedot_ye} shows the total electron capture and $\beta^-$ decay
rates for the two fixed temperature and density
calculations as a function of $Y_{\rm e}$ instead of time.
This figure shows how, as the material neutronizes, the total
electron capture rate declines and the total $\beta^-$ decay rate increases.
Once they meet, the rate of change of $Y_{\rm e}$ goes to zero and the system
reaches dWSE.

Four movies that accompany this paper show the evolution of the elemental
abundances in the four expansions as compared to those in
QSE, NSE, and dWSE.  Two other movies show the evolution of the
elemental abundances for the two fixed temperature and density calculations.
In the movies, it is clear that the nuclear populations evolve quickly
into QSE, then NSE, and finally, on a longer timescale (if the
expansion is not too fast), into dWSE.

Fig. \ref{fig:mass_tau_1e0} shows the mass fractions of the most abundant
species in the $\tau = 1$ s expansion.  This timescale is comparable to
the explosion timescales expected in Type Ia supernovae.
The dominant products are clearly
$^{66}$Ni, $^{60}$Fe, $^{50}$Ti, $^{48}$Ca, and $^{54}$Cr.  These five
species constitute nearly 86\% of the mass from this expanding matter.
About 14\% of the final mass is in $^{48}$Ca.  Expansions from lower
density do not reach as low a $Y_{\rm e}$ and, hence, produce much less $^{48}$Ca.
This result confirms that $^{48}$Ca production is robust when high densities
and timescales of seconds are present in expanding matter.

\begin{figure}[t]
\begin{center}
  \includegraphics[width=\figuresize\textwidth]{figures/wse_figures/mass_tau_1e0}
  \caption{Mass fractions as a function of $T_9$ in the $\tau = 1$ s expansion.}
\label{fig:mass_tau_1e0}
\end{center}
\end{figure}

\subsection{Conclusion}

Significant production of $^{48}$Ca in expansions of low-entropy matter
requires $Y_{\rm e} \approx 0.42$.  As we showed, densities near
$\rho = 9 \times
10^9$ g cm$^{-3}$ and expansion timescales of $\sim 1$ s can drive matter with
$Y_{\rm e} \approx 0.5$ down to this level and lead to robust $^{48}$Ca
synthesis.
This occurs despite the fact that such relatively fast expansions never
attain dynamical weak statistical equilibrium.

While Type Ia supernovae are fairly frequent events in the Galaxy and
significant producers of the Solar System's supply of iron,
a density as high as $9\times 10^9$ g cm$^{-3}$ is rare in white dwarf stars
(e.g., \cite{1997ApJ...476..801W}), the progenitors of these explosions.
This means that Type Ia supernova production of $^{48}$Ca is
probably rare.  Nevertheless, as our calculation shows, these rare events
would make tremendous amounts of $^{48}$Ca.  Because of this,
our expectation is that this
isotope is rather heterogeneously distributed in the dust in
the interstellar medium.  When such dust was inherited by the early
Solar nebula, the isotopically heterogeneous dust formed into solids
such as hibonites and FUN CAIs, which then naturally show isotopic anomalies
in the neutron-rich iron-group isotopes.


T. Y. gratefully acknowledges the support of a NASA Earth and Space Science
Fellowship.
 
This work is supported by NASA Grant NNX10AH78G.
